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We show that (i) if $q\\notin p\\mathbb{N}$ then there is unique Gibbs measure for any $k\\geq 1$ and $\\forall J_{xy}$ with $|J_{xy}|<p^{-1/(p-1)}$. (ii) For $q\\in p\\mathbb{N}, p\\geq 3$ one can choose $J_{xy}$ and $k\\geq 1$ such that there exist at least two Gibbs measures which are translat"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math-ph/0510024","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2005-10-06T14:12:22Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"ddb3d173a66eaba4e977d29e484b935243f5fafafe1af70a085cea9e337ea49d","abstract_canon_sha256":"6c3e70bfcfd9618854b41eb3a5f2d676f95ad9767c6f16a8e497ffd287fb3a20"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:35:38.685417Z","signature_b64":"FSZEdsuPyX6d+JuAYR1szsM5QCI0CHdWgSrP5FiS7IXWC6T9C7Gskv5USbOfSwnxH9JqfYtqlBKgXLw3TZrsCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f9b27c8b843ba4177141c19257a68b7a0bec64ff711bd320373ab2d5d0541494","last_reissued_at":"2026-05-18T02:35:38.685001Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:35:38.685001Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Inhomogeneous $p$-Adic Potts Model on a Cayley Tree","license":"","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Farrukh Mukhamedov, Utkir Rozikov","submitted_at":"2005-10-06T14:12:22Z","abstract_excerpt":"We consider a nearest-neighbor inhomogeneous $p$-adic Potts (with $q\\geq 2$ spin values) model on the Cayley tree of order $k\\geq 1$. 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(ii) For $q\\in p\\mathbb{N}, p\\geq 3$ one can choose $J_{xy}$ and $k\\geq 1$ such that there exist at least two Gibbs measures which are translat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0510024","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0510024","created_at":"2026-05-18T02:35:38.685055+00:00"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0510024v2","created_at":"2026-05-18T02:35:38.685055+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0510024","created_at":"2026-05-18T02:35:38.685055+00:00"},{"alias_kind":"pith_short_12","alias_value":"7GZHZC4EHOSB","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_16","alias_value":"7GZHZC4EHOSBO4KB","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_8","alias_value":"7GZHZC4E","created_at":"2026-05-18T12:25:53.335082+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI","json":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI.json","graph_json":"https://pith.science/api/pith-number/7GZHZC4EHOSBO4KBYGJFPJULPI/graph.json","events_json":"https://pith.science/api/pith-number/7GZHZC4EHOSBO4KBYGJFPJULPI/events.json","paper":"https://pith.science/paper/7GZHZC4E"},"agent_actions":{"view_html":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI","download_json":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI.json","view_paper":"https://pith.science/paper/7GZHZC4E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math-ph/0510024&json=true","fetch_graph":"https://pith.science/api/pith-number/7GZHZC4EHOSBO4KBYGJFPJULPI/graph.json","fetch_events":"https://pith.science/api/pith-number/7GZHZC4EHOSBO4KBYGJFPJULPI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/action/storage_attestation","attest_author":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/action/author_attestation","sign_citation":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/action/citation_signature","submit_replication":"https://pith.science/pith/7GZHZC4EHOSBO4KBYGJFPJULPI/action/replication_record"}},"created_at":"2026-05-18T02:35:38.685055+00:00","updated_at":"2026-05-18T02:35:38.685055+00:00"}